corresponding to the Particle Metropolis-Hastings Algorithm in [1]}.
Practical use of this algorithm requires tuning of the proposal distribution \(q(\tilde{\mathbf {\theta }}|\mathbf {\theta })\)  and of the number of particles used in each SMC run.
Concerning the latter, the number of particles in each SMC run needs to be reduced dramatically compared to a single run of the SMC algorithm since these runs have to be repeated many times. Although [1]} showed that the number of particles should increase with time, we have only considered a fixed number of particles in our testing. Even with many trials on the tuning parameters, we struggled to obtain convergence within reasonable time (running the algorithm for several days). However, we stored all the proposed parameter settings together with their likelihood values. We then fitted a two-dimensional GAM model to the log-likelihood values using \(a\)  and \(\sigma \)  as covariates. More than 500 samples of \((a,\sigma )\)  were obtained through multiple runs of the PMCMC algorithm with different starting points (running in total for several days). Figure REF  shows the fitted GAM curve with the red circles corresponding to the samples obtained from the SMC algorithm. The SMC samples seem to be within the higher likelihood region of the fitted curve. Note that due to lack of converge, the PMCMC algorithm has probably not been able to explore the full posterior properly. Further, the "observed" values displayed as small dots in the plot are all proposed values, not only the accepted ones, so these points should not be used to reflect the whole distribution.
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